Comparison of two semiclassical expansions for a family of PT-symmetric oscillators

Abstract

We show that a recently developed semiclassical expansion for the eigenvalues of PT-symmetric oscillators of the form $V(x)=(ix{)}^{2N+1}+bix$ does not agree with an earlier WKB expression for $V(x)=-(ix{)}^{2N+1}$ the case $b=0$. The reason is due to the choice of different paths in the complex plane for the calculation of the WKB integrals. We compare the Stokes and anti-Stokes lines that apply to each case for the quintic oscillator and derive a general WKB expression that contains the two earlier ones.